Group action invariants
| Degree $n$: | $45$ | |
| Transitive number $t$: | $26$ | |
| Group: | $C_3^2:(C_5:C_4)$ | |
| Parity: | $1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $1$ | |
| Generators: | (1,39,12,5,38,11,4,37,15,3,36,14,2,40,13)(6,29,17,10,28,16,9,27,20,8,26,19,7,30,18)(21,44,32,25,43,31,24,42,35,23,41,34,22,45,33), (1,36,6,16)(2,40,7,20)(3,39,8,19)(4,38,9,18)(5,37,10,17)(11,21,26,31)(12,25,27,35)(13,24,28,34)(14,23,29,33)(15,22,30,32)(42,45)(43,44) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $10$: $D_{5}$ $36$: $C_3^2:C_4$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 3: None
Degree 5: $D_{5}$
Degree 9: $C_3^2:C_4$
Degree 15: None
Low degree siblings
There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.
Conjugacy classes
| Cycle Type | Size | Order | Representative |
| $ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 6,41)( 7,42)( 8,43)( 9,44)(10,45)(11,36)(12,37)(13,38)(14,39)(15,40)(16,31) (17,32)(18,33)(19,34)(20,35)(21,26)(22,27)(23,28)(24,29)(25,30)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 1 $ | $45$ | $4$ | $( 2, 5)( 3, 4)( 6,21,41,26)( 7,25,42,30)( 8,24,43,29)( 9,23,44,28) (10,22,45,27)(11,16,36,31)(12,20,37,35)(13,19,38,34)(14,18,39,33)(15,17,40,32)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 1 $ | $45$ | $4$ | $( 2, 5)( 3, 4)( 6,26,41,21)( 7,30,42,25)( 8,29,43,24)( 9,28,44,23) (10,27,45,22)(11,31,36,16)(12,35,37,20)(13,34,38,19)(14,33,39,18)(15,32,40,17)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40) (41,42,43,44,45)$ |
| $ 10, 10, 10, 10, 5 $ | $18$ | $10$ | $( 1, 2, 3, 4, 5)( 6,42, 8,44,10,41, 7,43, 9,45)(11,37,13,39,15,36,12,38,14,40) (16,32,18,34,20,31,17,33,19,35)(21,27,23,29,25,26,22,28,24,30)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)(36,38,40,37,39) (41,43,45,42,44)$ |
| $ 10, 10, 10, 10, 5 $ | $18$ | $10$ | $( 1, 3, 5, 2, 4)( 6,43,10,42, 9,41, 8,45, 7,44)(11,38,15,37,14,36,13,40,12,39) (16,33,20,32,19,31,18,35,17,34)(21,28,25,27,24,26,23,30,22,29)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 6,41)( 2, 7,42)( 3, 8,43)( 4, 9,44)( 5,10,45)(11,16,21)(12,17,22) (13,18,23)(14,19,24)(15,20,25)(26,31,36)(27,32,37)(28,33,38)(29,34,39) (30,35,40)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1, 7,43, 4,10,41, 2, 8,44, 5, 6,42, 3, 9,45)(11,17,23,14,20,21,12,18,24,15, 16,22,13,19,25)(26,32,38,29,35,36,27,33,39,30,31,37,28,34,40)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1, 8,45, 2, 9,41, 3,10,42, 4, 6,43, 5, 7,44)(11,18,25,12,19,21,13,20,22,14, 16,23,15,17,24)(26,33,40,27,34,36,28,35,37,29,31,38,30,32,39)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1, 9,42, 5, 8,41, 4, 7,45, 3, 6,44, 2,10,43)(11,19,22,15,18,21,14,17,25,13, 16,24,12,20,23)(26,34,37,30,33,36,29,32,40,28,31,39,27,35,38)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1,10,44, 3, 7,41, 5, 9,43, 2, 6,45, 4, 8,42)(11,20,24,13,17,21,15,19,23,12, 16,25,14,18,22)(26,35,39,28,32,36,30,34,38,27,31,40,29,33,37)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,11,36)( 2,12,37)( 3,13,38)( 4,14,39)( 5,15,40)( 6,16,26)( 7,17,27) ( 8,18,28)( 9,19,29)(10,20,30)(21,31,41)(22,32,42)(23,33,43)(24,34,44) (25,35,45)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1,12,38, 4,15,36, 2,13,39, 5,11,37, 3,14,40)( 6,17,28, 9,20,26, 7,18,29,10, 16,27, 8,19,30)(21,32,43,24,35,41,22,33,44,25,31,42,23,34,45)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1,13,40, 2,14,36, 3,15,37, 4,11,38, 5,12,39)( 6,18,30, 7,19,26, 8,20,27, 9, 16,28,10,17,29)(21,33,45,22,34,41,23,35,42,24,31,43,25,32,44)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1,14,37, 5,13,36, 4,12,40, 3,11,39, 2,15,38)( 6,19,27,10,18,26, 9,17,30, 8, 16,29, 7,20,28)(21,34,42,25,33,41,24,32,45,23,31,44,22,35,43)$ |
| $ 15, 15, 15 $ | $4$ | $15$ | $( 1,15,39, 3,12,36, 5,14,38, 2,11,40, 4,13,37)( 6,20,29, 8,17,26,10,19,28, 7, 16,30, 9,18,27)(21,35,44,23,32,41,25,34,43,22,31,45,24,33,42)$ |
Group invariants
| Order: | $180=2^{2} \cdot 3^{2} \cdot 5$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [180, 24] |
| Character table: |
2 2 2 2 2 1 1 1 1 . . . . . . . . . .
3 2 . . . 2 . 2 . 2 2 2 2 2 2 2 2 2 2
5 1 1 . . 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1a 2a 4a 4b 5a 10a 5b 10b 3a 15a 15b 15c 15d 3b 15e 15f 15g 15h
2P 1a 1a 2a 2a 5b 5b 5a 5a 3a 15b 15d 15a 15c 3b 15f 15h 15e 15g
3P 1a 2a 4b 4a 5b 10b 5a 10a 1a 5b 5a 5a 5b 1a 5b 5a 5a 5b
5P 1a 2a 4a 4b 1a 2a 1a 2a 3a 3a 3a 3a 3a 3b 3b 3b 3b 3b
7P 1a 2a 4b 4a 5b 10b 5a 10a 3a 15b 15d 15a 15c 3b 15f 15h 15e 15g
11P 1a 2a 4b 4a 5a 10a 5b 10b 3a 15a 15b 15c 15d 3b 15e 15f 15g 15h
13P 1a 2a 4a 4b 5b 10b 5a 10a 3a 15c 15a 15d 15b 3b 15g 15e 15h 15f
X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
X.3 1 -1 A -A 1 -1 1 -1 1 1 1 1 1 1 1 1 1 1
X.4 1 -1 -A A 1 -1 1 -1 1 1 1 1 1 1 1 1 1 1
X.5 2 -2 . . B -B *B -*B 2 B *B *B B 2 B *B *B B
X.6 2 -2 . . *B -*B B -B 2 *B B B *B 2 *B B B *B
X.7 2 2 . . B B *B *B 2 B *B *B B 2 B *B *B B
X.8 2 2 . . *B *B B B 2 *B B B *B 2 *B B B *B
X.9 4 . . . 4 . 4 . -2 -2 -2 -2 -2 1 1 1 1 1
X.10 4 . . . 4 . 4 . 1 1 1 1 1 -2 -2 -2 -2 -2
X.11 4 . . . C . *C . -2 -B -*B -*B -B 1 D /E E /D
X.12 4 . . . C . *C . -2 -B -*B -*B -B 1 /D E /E D
X.13 4 . . . *C . C . -2 -*B -B -B -*B 1 E D /D /E
X.14 4 . . . *C . C . -2 -*B -B -B -*B 1 /E /D D E
X.15 4 . . . C . *C . 1 D /E E /D -2 -B -*B -*B -B
X.16 4 . . . C . *C . 1 /D E /E D -2 -B -*B -*B -B
X.17 4 . . . *C . C . 1 E D /D /E -2 -*B -B -B -*B
X.18 4 . . . *C . C . 1 /E /D D E -2 -*B -B -B -*B
A = -E(4)
= -Sqrt(-1) = -i
B = E(5)^2+E(5)^3
= (-1-Sqrt(5))/2 = -1-b5
C = 2*E(5)^2+2*E(5)^3
= -1-Sqrt(5) = -1-r5
D = -E(5)^2+2*E(5)^3
E = -E(5)+2*E(5)^4
|